Abstract
We study the asymptotic behaviour of random walks in i.i.d. non-elliptic random environments on $\mathbb{Z} ^d$. Standard conditions for ballisticity and the central limit theorem require ellipticity, and are typically non-local. We use oriented percolation and martingale arguments to find non-trivial local conditions for ballisticity and an annealed invariance principle in the non-elliptic setting. The use of percolation allows certain non-elliptic models to be treated even though ballisticity has not been proved for elliptic perturbations of these models.
Highlights
A central topic in modern probability and statistical physics is the study of random walks in random media
Existing results have largely been restricted to situations where the random environment is elliptic, i.e. where steps to all nearest neighbours are possible
Random walks in i.i.d. random environments are not reversible
Summary
A central topic in modern probability and statistical physics is the study of random walks in random media. We will study random walks in i.i.d. random environments (RWRE) that are non-elliptic, such as in the following example. Dimensional-orthant model (Example 1) is the uniform RWRE on the random directed graph which has Gx = ←↓ with probability 1 − p. In many settings we will be able to conclude that the range of the walker satisfies condition (b) of the theorem by proving that it holds for the range of such a transverse walk. We will find that in Example 1, having strong barriers ←↓ is an insurmountable obstacle to obtaining a positive speed in direction l = (1, 1) using one of the standard ballisticity conditions. One way of interpreting our results in the context of Example 1 is that we can overcome the presence of strong barriers ←↓ by strengthening the forward push and including sufficiently many sites ↑→ that don’t permit backwards motion
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